When I tell a friend I'm off to meet Britain's best-known same-sex couple and their five children, he says casually: "They're just another set of parents. OK, so five kids is a lot – but really, so what?" Which may be a measure of how far we have come in accepting different versions of the family.

It was very different a decade ago when Barrie and Tony Drewitt-Barlow first made the headlines after they fought – and won – a battle to bring their twins, conceived via donor eggs and carried to term by a surrogate mother, home to Britain after they were born in America. When Eamonn Holmes interviewed them in 2000 after bringing Saffron and Aspen back to the UK, he told them: "There are people who are going to say it's not natural. [They] will say you've been shopping for the ultimate gay accessory."

Today, things have moved on in the Drewitt-Barlow household. Tony, now 45, and Barrie, 41, arrive at our meeting with not only the twins, now 10, but their three siblings as well – Orlando, six, and twins Jasper and Dallas, five months – all conceived, like their elder brother and sister, with donor eggs and surrogate mothers.

There is a lot more acceptance of the couple and their family now than there was a decade ago, says Barrie, who turns out to be the chattier of the two. "Ten years ago everyone thought we were a pair of paedophiles. Gay men wanting to be parents! What else could we be? We had to be paedophiles.

"We've spent the last 10 years trying to persuade people that's not the case. We've spent 10 years getting people to understand we're just another family."

On the other hand, the men recently removed their children from school after what they describe as a "campaign" against their family. "We were bullied. The children weren't bullied, but Tony and I were – by homophobic parents."

So it seems not everyone is comfortable with the Drewitt-Barlows, who appear larger than life in every direction. It's not just that they have kids, they have five kids; it's not just that they are rich, they are millionaires. So the surprising thing when you meet them is how ordinary they are in many respects. When you discuss the day-to-day detail of family life, they sound like any long-term married, big-family couple, feeling the strain of two new babies plus three older kids. Ask them how it's going with five rather than three children and they give you the same sort of answer any couple would. "It's put a strain on our relationship, but we expected that," says Barrie. "We notice the difference being older – we're knackered a lot of the time. And Tony and I don't get much time on our own together these days."

Like many couples, Barrie and Tony have clearly defined roles. Barrie is the nurturer; he says he gets broody and it was his idea to have Jasper and Dallas. "I do the cooking, I do the ironing. I organise the playdates, I take the children to school. I work out what's going to happen when Saffron's rounders game at one end of the county clashes with Aspen's cricket match in the other direction, and they both need picking up at the exact same time that the nanny finishes. I'm the family organiser, Tony is the breadwinner."

As if on cue, Tony mutters something about having to get back to his desk – it's Friday afternoon, and he's got a lot to finish before he can start the weekend. Tony has his work cut out running a family with the overheads of this one: the three older children are at an independent school, Saffron has a pony and all the holidays mentioned in passing tend to be of the Caribbean or Florida variety. All this on top of the initial price tag of getting their children in the first place: neither Tony nor Barrie has a figure for what their children cost them pre-birth, but both admit it was a lot, and some newspapers have speculated that, with various egg donors and surrogate mothers to pay, hospital bills, plus innumerable trips back and forth across the Atlantic, it came to at least £1m. None of it would have been possible were it not for the success of their clinical testing company, which they sold in 1998 for a reported £4m.

But there has been a bit of trouble recently over their business dealings. Last week, it emerged that the pair had mishandled a company wind-up, moving money so that its creditors lost out. They have each accepted an eight-year government ban on holding any directorships. But they didn't want to discuss this with me.

What Tony's fed up with, he says, is the assumption that because they are well-off everything is easy. They do have a nanny from 8am to 6pm weekdays. "But at 6pm she goes home and then it's just us. So on a Friday night I'll get in from work and it's literally straight into looking after the babies. There's no respite: the weekend is heavy domestic duty and then on Monday morning I'm back into work from first thing." What he would most like, he says, is for people to realise how normal their lives are. "It's not about spoilt children and designer shoes, the way some people seem to think. We're just an ordinary family, with an ordinary kitchen, an ordinary garden and ordinary goings-on."

But not everything is ordinary in the Drewitt-Barlow household. Not many families are made up of children who are not – and possibly never will be – sure who their genetic father is. "The children were conceived using donor eggs and our sperm," explains Barrie. "We know whose child is whose, but we haven't told anyone else – not even the children. As far as we're concerned, we're their parents. It's that simple. They don't need to know whether they're Tony's or mine – they all belong to both of us." But surely every child has the right to know – if at all possible – who their genetic parents are? What if, as they get older, they want to know? "I'm not saying it's never going to happen, and if it does we'll deal with it."

In a new Channel 4 film on the Drewitt-Barlow family, the children are seen discussing which traits are from Dad (Tony) and which from Daddy (Barrie). "I've got Daddy's nose and Dad's eyes," says Aspen. "I've got Dad's brain, and Daddy's eyes," Saffron replies.

Then there's the fact that the Drewitt-Barlow extended family embraces various individuals whose involvement was crucial for the children's existence – such as Rosalind Bellamy, who is over from her home in California at the moment (there's just been a big family gathering for the twins' christening), staying with the family in their Essex village home. She joins us for the interview. Rosalind was Aspen and Saffron's surrogate mother. It's lovely to see them from time to time, she says, but no – she doesn't feel a maternal link. In fact, the babies weren't genetically hers – the egg donor was Tracey, with whom the Drewitt-Barlows seem to have a frostier relationship, although she does chat to the children sometimes on Skype.

Tracey was also the egg donor for Orlando, but he was carried by another surrogate mother, Donna; and Donna also gave birth to the new twins, Jasper and Dallas, although they had a different egg donor, a 6ft-tall model called Andrea (who will not, says Barrie, play any role in their lives).

It's all mind-bogglingly complex. "We have trouble getting our heads round it ourselves sometimes," says Tony, as I struggle to get the details right in my notebook. But the important thing, they maintain, is that it works just fine for them. "We wanted children, and there wasn't another way to do it," says Tony. Why did they want so many children? "We're both from big families ourselves – Tony is one of four and so am I, and my mum was a foster carer as well," says Barrie. "We loved the idea of a big family."

In fact, parenthood was already beckoning when the pair met in 1988. At the time, Barrie had just – unbeknown to him – fathered a son, Colin, who was to come into his life many years later and with whom he is still in regular contact; Tony was about to get married and would almost certainly have gone on to become a conventional dad. "I was a closet gay – I was terrified of coming out, of what it would mean," says Tony. "The irony is that when I told my mum and dad, they couldn't have been better about it."

Within a short while, the wedding was cancelled and Tony and Barrie had forged their own partnership, which has lasted to this day. It was to be 10 years, though, before they decided to start the quest to have children and those child-free years, says Barrie, were to strengthen them as a couple, and have helped them to cope.

They are not just talking about the prejudice they have met as gay fathers. In 2006, Tony was diagnosed with cancer of the mouth. "It was a complete shock – I'm not a smoker," he says. The tumour was too advanced for surgery, so he had to cope with many weeks of radiotherapy. His weight dropped to six stone. Relating the story, both men seem shell-shocked at the memory. "I didn't want to be a single parent," says Barrie, in a voice quieter than usual. Tony just shakes his head and says Barrie went "to hell and back" over his illness, and that if it ever returned he would rather not know than have to go through all that again.

Tony's illness is in remission but still it seems a giant leap of faith to have had two more babies. "We do worry about the future," Tony admits. "But every parent worries about the future – there's no end to the worrying once you've got kids. And this was about tenacity, about believing in the future." Barrie, meanwhile, says he's "not even thinking about the cancer any more because I know Tony is fine".

What of the most important people in this story, the children – Saffron, Aspen, Orlando, Jasper and Dallas? Not only is their family set-up more complex and unconventional than most, they are also growing up in the glare of publicity. Barrie says they do sometimes worry about the exposure the children have had, though he seems oddly unable to link it to anything he might be doing to court it. While we've been chatting, the babies have been gurgling happily, Aspen and Orlando have been doodling on paper, and Saffron has been listening intently.

What is it like for her, I ask, the lone female in this all-male set up? Didn't she long for the new babies (whose sex was left to chance) to be girls? Saffron admits that she did but, not wanting to be disloyal to her new little brothers, immediately adds that she's completely delighted with them.

"I'd rather have a dad and a daddy than a mum and a dad," says Aspen in the film. "But you can't make babies like us in Britain – it's not allowed. I think that sucks." And: "If they want to have children, they should be allowed to."

Do Tony and Barrie mind whether their kids turn out to be gay or straight? "We don't want them to feel they're offending us by not being gay," says Barrie. Tony, meanwhile, says that in the past he'd have hoped the children were straight because it would be so much easier for them growing up. "But things are changing, so I wouldn't mind so much these days," he says.

"Now, I'd say the only important thing is that I don't want any of them to have to live a lie. That's a terrible thing. I want our children to be themselves; it's the most important thing to be."

Cutting Edge: My Weird & Wonderful Family will be shown on Channel 4 on 21 July at 9pm